Graphing F(x) = -1/(x-6): Domain, Intercepts, Asymptotes

by Admin 57 views
Graphing f(x) = -1/(x-6): Domain, Intercepts, Asymptotes

Let's break down how to analyze and graph the function f(x) = -1/(x-6). We'll cover finding the domain, intercepts, horizontal asymptotes, and then put it all together to sketch the graph. This is a rational function, so understanding these key features is super important.

Finding the Domain

When we talk about the domain, we're asking: what are all the possible x values that we can plug into the function without causing any mathematical mayhem? In other words, what values of x will give us a real number output? For rational functions like this one, the main thing we need to watch out for is division by zero. Division by zero is undefined, and it'll break our function. So, we need to figure out what x value makes the denominator equal to zero.

Looking at our function, f(x) = -1/(x-6), the denominator is (x-6). We need to find the value of x that makes (x-6) = 0. A simple bit of algebra tells us that x = 6 is the culprit. If we plug in x = 6, we get -1/(6-6) = -1/0, which is a big no-no in the math world.

Therefore, the domain is all real numbers except for 6. We can write this in a few different ways:

  • Set notation: { x | x ≠ 6 }
  • Interval notation: (-∞, 6) ∪ (6, ∞)

Understanding the Domain is Key: Grasping the domain is fundamental because it dictates the landscape on which our function exists. For f(x) = -1/(x-6), the domain excludes x = 6, creating a vertical asymptote that significantly influences the graph's behavior. This exclusion is crucial for accurately visualizing and interpreting the function's properties.

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, like the one we're dealing with, the domain is restricted by any values that would make the denominator equal to zero. Because division by zero is undefined, these values must be excluded from the domain. In our function, f(x) = -1/(x - 6), the denominator is (x - 6). To find the values that make the denominator zero, we set (x - 6) = 0. Solving this equation gives us x = 6. This means that x = 6 is the value we must exclude from the domain. Therefore, the domain of f(x) is all real numbers except x = 6. In interval notation, this is represented as (-∞, 6) ∪ (6, ∞), indicating that the function is defined for all real numbers less than 6 and all real numbers greater than 6, but not at 6 itself. This concept is crucial because it directly relates to the existence of a vertical asymptote at x = 6, which we will discuss later. Recognizing and correctly determining the domain is the first step in understanding and graphing rational functions.

Finding the Intercepts

Next up, let's find where our function crosses the x and y axes. These points are called the intercepts, and they give us important anchor points when we're sketching the graph.

Y-intercept

The y-intercept is the point where the graph crosses the y-axis. This happens when x = 0. So, to find the y-intercept, we simply plug in x = 0 into our function:

  • f(0) = -1/(0 - 6) = -1/(-6) = 1/6

So, the y-intercept is the point (0, 1/6). That's a pretty straightforward calculation, guys!

X-intercept

The x-intercept is where the graph crosses the x-axis. This is where f(x) = 0 (or y = 0). So, we need to solve the equation:

  • 0 = -1/(x - 6)

Now, this is where things get interesting. For a fraction to be equal to zero, the numerator has to be zero. Our numerator is -1, which is never zero. This means there's no value of x that will make the function equal to zero. Therefore, there is no x-intercept. This is a key piece of information, and it tells us the graph will never touch the x-axis.

Significance of Intercepts: The intercepts offer essential reference points for plotting the graph. The y-intercept (0, 1/6) anchors the graph's position relative to the y-axis, while the absence of an x-intercept indicates that the function does not intersect the x-axis. This non-intersection is a critical feature that shapes the graph's overall form and behavior.

To find the intercepts, we look for the points where the graph of the function intersects the x-axis (x-intercepts) and the y-axis (y-intercept). The y-intercept is the point where x = 0. To find it, we substitute x = 0 into the function: f(0) = -1/(0 - 6) = -1/(-6) = 1/6. Therefore, the y-intercept is the point (0, 1/6). For the x-intercepts, we look for the points where f(x) = 0. We set the function equal to zero and solve for x: 0 = -1/(x - 6). For a fraction to be zero, its numerator must be zero. However, in this case, the numerator is -1, which is never zero. This means there is no value of x that will make the function equal to zero. Consequently, there are no x-intercepts for this function. The absence of x-intercepts tells us that the graph of the function will never cross the x-axis. This is an important characteristic to note when we are sketching the graph, as it provides a clear boundary that the graph will not cross.

Finding Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x gets really, really big (approaches positive infinity) or really, really small (approaches negative infinity). In other words, what value does f(x) approach as x goes way out to the sides?

To find horizontal asymptotes, we compare the degrees of the numerator and denominator. The degree of a polynomial is the highest power of x. In our case:

  • The numerator is -1, which can be thought of as -1x⁰. So, the degree of the numerator is 0.
  • The denominator is (x - 6), which has a degree of 1 (because the highest power of x is x¹).

Here's the rule to remember:

  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. (This is our case!)
  • If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator).
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (but there might be a slant asymptote).

Since the degree of the numerator (0) is less than the degree of the denominator (1), we have a horizontal asymptote at y = 0. This means as x gets super large (positive or negative), the function's value gets closer and closer to 0, but it never actually reaches it.

Horizontal Asymptotes in Graphing: The horizontal asymptote y = 0 serves as a guide for the graph's end behavior, illustrating that as x moves towards positive or negative infinity, the function values approach zero without ever reaching it. This is crucial for understanding the overall trend and limitations of the function's output.

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x approaches positive or negative infinity. To find horizontal asymptotes, we compare the degrees of the polynomials in the numerator and the denominator. The degree of a polynomial is the highest power of x in the polynomial. In our function, f(x) = -1/(x - 6), the numerator is -1, which can be thought of as a constant polynomial with a degree of 0. The denominator is (x - 6), which is a linear polynomial with a degree of 1. When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. This is because as x becomes very large (either positively or negatively), the denominator grows much faster than the numerator, causing the fraction to approach zero. Therefore, the horizontal asymptote for f(x) = -1/(x - 6) is y = 0. This means that as x gets larger and larger in either the positive or negative direction, the graph of the function will get closer and closer to the line y = 0, but it will never actually cross it unless there are specific conditions that allow it to.

Graphing the Function

Okay, guys, we've got all the pieces of the puzzle! Let's put it together to graph f(x) = -1/(x-6).

  1. Vertical Asymptote: We know there's a vertical asymptote at x = 6. Draw a dashed vertical line at x = 6. This line is a boundary the graph can't cross.
  2. Horizontal Asymptote: We found a horizontal asymptote at y = 0. Draw a dashed horizontal line along the x-axis (since y = 0 is the x-axis). Again, this is a line the graph will approach but not cross (in most cases).
  3. Y-intercept: We plotted the y-intercept at (0, 1/6). This is one point we know for sure.
  4. No X-intercept: We know the graph doesn't cross the x-axis.
  5. Test Points: To get a better sense of the shape, let's pick some test points. Choose x values on either side of the vertical asymptote:
    • x = 5: f(5) = -1/(5 - 6) = -1/(-1) = 1. So, we have the point (5, 1).
    • x = 7: f(7) = -1/(7 - 6) = -1/1 = -1. So, we have the point (7, -1).
  6. Sketch the Graph: Now, we can sketch the graph. Remember:
    • The graph will approach the asymptotes but never cross them (unless there's a specific reason, which there isn't in this case).
    • The graph will pass through the points we plotted.
    • The negative sign in front of the 1 in our function f(x) = -1/(x - 6) means the graph will be reflected across the x-axis compared to the basic function 1/(x-6).

You'll see the graph has two separate curves. One curve is in the top-left quadrant (above the x-axis and to the left of the vertical asymptote), and the other curve is in the bottom-right quadrant (below the x-axis and to the right of the vertical asymptote). They both hug the asymptotes.

Visualizing the Graph: The graph showcases how the function behaves around its asymptotes and intercepts. The vertical asymptote at x = 6 divides the graph into two sections, each approaching the horizontal asymptote y = 0 as x moves towards infinity. The y-intercept (0, 1/6) provides a specific point of reference, while the absence of an x-intercept confirms that the graph never crosses the x-axis. These elements combined give a clear picture of the function's shape and characteristics.

Putting It All Together to Graph: To graph f(x) = -1/(x - 6), we start by drawing the asymptotes as dashed lines. The vertical asymptote is at x = 6, and the horizontal asymptote is at y = 0 (the x-axis). We then plot the y-intercept, which is (0, 1/6). Since there are no x-intercepts, the graph will not cross the x-axis. To get a better sense of the shape of the graph, we choose test points on either side of the vertical asymptote. For example, let’s try x = 5 and x = 7. When x = 5, f(5) = -1/(5 - 6) = 1. This gives us the point (5, 1). When x = 7, f(7) = -1/(7 - 6) = -1. This gives us the point (7, -1). With these points and the knowledge of the asymptotes, we can sketch the graph. The graph will have two branches. One branch will be in the second quadrant, approaching the asymptotes x = 6 and y = 0. The other branch will be in the fourth quadrant, also approaching the asymptotes. The negative sign in the function f(x) = -1/(x - 6) means that the graph is reflected across the x-axis compared to the graph of f(x) = 1/(x - 6). The branches will get closer and closer to the asymptotes as x approaches infinity or 6, but they will never actually touch or cross the asymptotes.

In Summary

To analyze and graph a rational function like f(x) = -1/(x-6), we follow these steps:

  1. Find the domain by identifying any values of x that make the denominator zero.
  2. Find the intercepts by setting x = 0 for the y-intercept and f(x) = 0 for the x-intercept.
  3. Find horizontal asymptotes by comparing the degrees of the numerator and denominator.
  4. Identify vertical asymptotes where the denominator equals zero.
  5. Plot key points and sketch the graph, using the asymptotes as guides.

By following these steps, you can confidently analyze and graph rational functions, guys! Remember to pay close attention to those asymptotes – they're the key to understanding the function's behavior.